Exact Control Of A Kind Nonlinear Evolution Equation

Exact controllability is one kind of distributed parameter controls, which has been emphasized in the Control Theory and has been extensively studied and developed. Recently people more and more take notice on the exact control of Burgers equation, KdV equation, and K-S equation.Firstly,This paper studies the problem of the exact controllability for the Burgers-Kdv equation.Using the Leray-Schauder fixed point theorem and the theory of dissipative operators, we first consider the solution of the linear Burgers-Kdv equation. Second we find the control function h by using the theory of monotone operators and division integral theory .Last we prove the exact controllability for the Burgers-Kdv equation.Secondly,we consider the exact boundary control of a generalised Burgers-Kdv equation posedd on a finite interval x∈[α,β],t∈[0,T].By using the theory of semigroup operators, we first consider the exat control described by the linear generalised Burgers-Kdv equation,with fixed point theorem and the theory of Fredholm we proved the system is exactly controled. Then by considering initial value control of the generalised Burgers-Kdv equation posed on the whole real line R we prove the exact boundary controllability for the generalised Burgers-Kdv equation.